Efficient sympathetic motional-ground-state cooling of a molecular ion

Cold molecular ions are promising candidates in various fields ranging from precision spectroscopy and test of fundamental physics to ultracold chemistry. Control of internal and external degrees of freedom is a prerequisite for many of these applications. Motional-ground-state cooling represents the starting point for quantum logic-assisted internal state preparation, detection, and spectroscopy protocols. Robust and fast cooling is crucial to maximize the fraction of time available for the actual experiment. We optimize the cooling rate of ground-state cooling schemes for single ${}^{25}{\mathrm{Mg}}^{+}$ ions and sympathetic ground-state cooling of ${}^{24}{\mathrm{MgH}}^{+}$. In particular, we show that robust cooling is achieved by combining pulsed Raman sideband cooling with continuous quench cooling. Furthermore, we experimentally demonstrate an efficient strategy for ground-state cooling outside the Lamb-Dicke regime.

Figure 1

Principle of resolved sideband cooling and implementation in ${}^{25}{\mathrm{Mg}}^{+}$. (a) Resolved SBC requires the linewidth $\Gamma$ of the transition to be smaller than the trap frequency ${\omega }_{\mathrm{T}}$, so that individual transitions can be selectively driven. With the cooling laser tuned to the first-order RSB, one quantum of motion is removed from the system in each absorption event. In the Lamb-Dicke regime, a change in motional state is suppressed upon decay back to the $|\downarrow \rangle$ state. (b) Relevant level structure for SBC of a ${}^{25}{\mathrm{Mg}}^{+}$ ion. Doppler cooling and Raman transitions are performed by coupling the ${}^2{S}_{1/2}$ and ${}^2{P}_{3/2}$ states. Compared to a previous implementation of SBC [67], a second laser is added for repumping the $|\mathrm{aux}\rangle$ state via the ${}^2{S}_{1/2}\leftrightarrow {}^2{P}_{1/2}$ transition. rf radiation couples the states $|\uparrow \rangle$ and $|\mathrm{aux}\rangle$.

Figure 2

Motional state population and effective Rabi frequencies. (a) Typical population distribution over motional states after Doppler cooling, corresponding to $\overline{n}\approx 10$. (b) Effective Rabi frequencies as a function of the motional state calculated with a Lamb-Dicke parameter $\eta =0.30$ according to Eq. (1). The effective Rabi frequencies for different RSB orders have zero points at different motional excitations. Solid/dashed/dotted: first/second/third-order RSB.

Figure 3

Sequence for single ion sideband cooling. Sequence for SBC a single ${}^{25}{\mathrm{Mg}}^{+}$. The sequence starts by repeating $N_{\mathrm{R}}2$ second-order RSBs, followed by $N_{\mathrm{R}}1$ first-order RSBs with fixed pulse lengths. After each SBC pulse, repumping pulses consisting of ${}^2{P}_{1/2}$ and $\sigma$ beams are applied to bring the ion back to $|\downarrow \rangle$ state.

Figure 4

Dependence of the cooling time constants on experimental parameters. Cooling time constants as a function of the pulse lengths of the (a) second-order and (b) first-order RSB pulses with $\alpha =0.5$ while fixing the nonscanned pulse length near its optimal value. Blue point (red square): Experimentally determined cooling constants with the quench coupling on (off) during the RSB pulse. The quench coupling induces an effective decay rate of $42{\mathrm{ms}}^{-1}$ from the $|\uparrow \rangle$ state to the $|\downarrow \rangle$ state. (c) Cooling time constant as a function of the time scaling factor with $t_{\mathrm{R}}2=10\mu \mathrm{s}$ and $t_{\mathrm{R}}1=10\mu \mathrm{s}$. The lines in (a)–(c) are the result of master equation simulations using the experimental parameters. (d) Residual motional excitation as a function of the total cooling time $T_{\mathrm{c}}$. The line is the fit of Eq. (5) to the experimental data, which gives the cooling constant $T_0$.

Figure 5

Rabi excitation of ${}^{25}{\mathrm{Mg}}^{+}$ after sideband cooling. (a)–(c) Frequency scans of Raman transitions over first RSB (red), carrier (black), and first BSB (blue) transitions after SBC of a single ${}^{25}{\mathrm{Mg}}^{+}$.

Figure 6

Sequence for sympathetic sideband cooling of a two-ion crystal. The in-phase and out-of-phase axial modes of a ${}^{25}{\mathrm{Mg}}^{+}/{}^{24}{\mathrm{MgH}}^{+}$ two-ion crystal are cooled in an interleaved fashion to ensure simultaneous cooling.

Figure 7

Dependence of the cooling time constant on experimental parameters for sympathetic sideband cooling a two-ion crystal. (a) The cooling time constant as a function of the optimal pulse length of the Raman RSB pulses on the IP (blue circles) and OP (red squares) mode. (b) The cooling constant as a function of the optimal time scaling factor ${\alpha }^{\prime }$. The lines connect the points and are guides to the eye.

Figure 8

Rabi excitation of ${}^{25}{\mathrm{Mg}}^{+}$ after two-mode SBC a ${}^{24}{\mathrm{MgH}}^{+}/{}^{25}{\mathrm{Mg}}^{+}$ crystal. Frequency scans of Raman transitions over the first RSB (red) of the (a) OP and (b) IP modes, (c) the carrier (black), and the first BSB (blue) (d) IP and (e) OP modes after SBC both modes on ${}^{25}{\mathrm{Mg}}^{+}$.

Figure 9

Motional state population and effective Rabi frequency for small trap frequency. (a) Distribution of the motional levels at a motional frequency of ${\omega }_{\mathrm{T}}=1\mathrm{M}\mathrm{Hz}$ and a temperature of $T=1\mathrm{mK}$ corresponding to the Doppler cooling limit of the ${}^{25}{\mathrm{Mg}}^{+}$ ion. Less than 0.3% of the population are in the trap levels $|n>120\rangle$. (b) Effective Rabi frequencies at $\eta =0.45$ for different RSB orders are shown as a function of the trap levels. For ions located in higher trap levels, RSB pulses of higher-orders $\beta \left(n\right)$ are more efficient for cooling.

Figure 10

Determination of the maximum sideband order. (a) The cooling time constant as a function of applied sideband orders. The optimal number for the sideband orders determined experimentally confirms the prediction shown in Fig. 9. (b) RSB orders with highest effective cooling rate as a function of the trap level and the Lamb-Dicke parameter. With known Lamb-Dicke parameter and motional distribution a corresponding SBC sequence can be adopted.

Figure 11

Relevant levels for numerical simulation of the dynamics during the Raman SBC cycles. The Raman lasers are tuned to be resonant with the $\beta$-order RSB ($\beta =1$ in graph). Through the dissipative channels either from off-resonant excitation or via a repumping process followed by spontaneous emission, the ion is reinitialized to the $|\downarrow \rangle$ with a decay rate of 0.005 and $0.47{\mathrm{s}}^{-1}$, respectively. Spontaneous emission happens on the carrier transition with a probability $1-\xi$ and the remaining small fraction $\xi$ happens on the RSB transitions.